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Journal Article

The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra


Huang,  Yi-Zhi
Max Planck Institute for Mathematics, Max Planck Society;

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Huang, Y.-Z., & Qi, F. (2020). The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra. Transactions of the American Mathematical Society, 373(11), 7817-7868. doi:10.1090/tran/8240.

Cite as: http://hdl.handle.net/21.11116/0000-0007-5F3A-F
We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$ elements such that the difference between the skew-symmetric opposite action of the left action and the right action on these elements are Laurent polynomials in the variable. If $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ for every $\Z$-graded $V$-bimodule $M$, then every left $V$-module satisfying a composability condition is completely reducible. In particular, since a lower-bounded $\Z$-graded vertex algebra $V$ is a special meromorphic open-string vertex algebra and left $V$-modules are in fact what has been called generalized $V$-modules with lower-bounded weights (or lower-bounded generalized $V$-modules), this result provides a cohomological criterion for the complete reducibility of lower-bounded generalized modules for such a vertex algebra. We conjecture that the converse of the main theorem above is also true. We also prove that when a grading-restricted vertex algebra $V$ contains a subalgebra satisfying some familiar conditions, the composability condition for grading-restricted generalized $V$-modules always holds and we need $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ only for every $\Z$-graded $V$-bimodule $M$ generated by a grading-restricted subspace in our complete reducibility theorem.